Question #a927e

2 Answers

We know

cos2theta=cos^2theta-sin^2theta

cos2theta=cos^2theta-(1-cos^2theta)

cos2theta=2cos^2theta-1

cos^2theta=1/2*(1+cos2theta)

costheta=sqrt(1/2(1+cos2*theta))

putting theta=22.5 we get

cos(22.5)=sqrt(1/2(1+cos45^o)

cos(22.5)=sqrt(1/2(1+1/sqrt2)

Mar 12, 2017

sqrt(2 + sqrt2)/2

Explanation:

Call cos (22.5) = cos t
cos 2t = cos (45) = sqrt2/2
Use trig identity:
2cos^2 t = 1 + cos 2t = 1 + sqrt2/2 = (2 + sqrt2)/2
cos^2 t = (2 + sqrt2)/4
cos t = +- sqrt(2 + sqrt2)/2
Since cos 22.5 is positive, then, take the positive answer.
cos (22.5) = cos t = sqrt(2 +sqrt2)/2